Olympiad problems

2007 All-Russian Olympiad, 1

Prove that for $k>10$ Nazar may replace in the following product some one $\cos$ by $\sin$ so that the new function $f_{1}(x)$ would satisfy inequality $|f_{1}(x)|\le 3\cdot 2^{-1-k}$ for all real $x$. \[f(x) = \cos x \cos 2x \cos 3x \dots \cos 2^{k}x \] [i]N. Agakhanov[/i]

1980 IMO, 23

Tags: complex numbers , inequalities unsolved , inequalities

Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of \[\left|\frac{x + y}{1 + x\overline{y}}\right|\]

1996 APMO, 2

Tags: inequalities , inequalities unsolved

Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that \[ 2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \]

2009 South africa National Olympiad, 4

Tags: inequalities , inequalities unsolved

Let $x_1,x_2,\dots,x_n$ be a finite sequence of real numbersm mwhere $0<x_i<1$ for all $i=1,2,\dots,n$. Put $P=x_1x_2\cdots x_n$, $S=x_1+x_2+\cdots+x_n$ and $T=\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}$. Prove that \[\frac{T-S}{1-P}>2.\]

2003 Mediterranean Mathematics Olympiad, 3

Tags: inequalities , blogs , inequalities unsolved , algebra , highschoolmath , High school olympiad

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

2007 Greece National Olympiad, 2

Tags: inequalities , inequalities unsolved

Let $a,b,c$ be sides of a triangle, show that \[\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.\]

2011 IFYM, Sozopol, 2

Tags: inequalities , inequalities unsolved

prove that $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) \leq 1$ for $0 < a < b \leq c < d$ and when $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) = 1 $

2014 Iran Team Selection Test, 5

Tags: inequalities , function , inequalities unsolved , n-variable inequality

$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that: $\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$

2005 IMAR Test, 2

Tags: function , inequalities unsolved , inequalities

Let $n \geq 3$ be an integer and let $a,b\in\mathbb{R}$ such that $nb\geq a^2$. We consider the set \[ X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . \] Find the image of the function $M: X\to \mathbb{R}$ given by \[ M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . \] [i]Dan Schwarz[/i]

1984 IMO Longlists, 29

Tags: inequalities , function , induction , inequalities unsolved

Let $S_n = \{1, \cdots, n\}$ and let $f$ be a function that maps every subset of $S_n$ into a positive real number and satisfies the following condition: For all $A \subseteq S_n$ and $x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)$. Prove that for all $A,B \subseteq S_n$ the following inequality holds: \[f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)\]

Oliforum Contest I 2008, 1

Tags: inequalities unsolved , inequalities

Let $ a,b,c$ positive reals such that $ ab \plus{} bc \plus{} ca \equal{} 3$, show that: $ \displaystyle a^2 \plus{} b^2 \plus{} c^2 \plus{} 3 \ge \frac {a(3 \plus{} bc)^2}{(c \plus{} b)(b^2 \plus{} 3)} \plus{} \frac {b(3 \plus{} ca)^2}{(a \plus{} c)(c^2 \plus{} 3)} \plus{} \frac {c(3 \plus{} ab)^2}{(b \plus{} a)(a^2 \plus{} 3)}$ ([i]Anass BenTaleb, Ali Ben Bari High School - Taza,Morocco[/i])

2005 MOP Homework, 4

Tags: inequalities unsolved , inequalities

Let $x_1$, $x_2$, ..., $x_5$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5=5$. Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.

2009 Korea - Final Round, 1

Tags: inequalities , inequalities unsolved

$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$, $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$$+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$. Prove that $AB \ge 9$.

1972 IMO Longlists, 25

Tags: inequalities , inequalities unsolved

We consider $n$ real variables $x_i(1 \le i \le n)$, where $n$ is an integer and $n \ge 2$. The product of these variables will be denoted by $p$, their sum by $s$, and the sum of their squares by $S$. Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$. Prove that it holds for every $n$-tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$

1983 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , inequalities unsolved , inequalities

Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n\plus{}1}\minus{}1)$ in base $ 10$ are $ 28$.

1972 IMO Longlists, 9

Tags: inequalities unsolved , inequalities

Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.

2003 China Team Selection Test, 1

Tags: trigonometry , inequalities unsolved , inequalities

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

2014 South East Mathematical Olympiad, 5

Tags: inequalities , inequalities unsolved

Let $x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1+x_2+\cdots+x_n=1$ $(n\ge 2)$. Prove that\[\sum_{i=1}^n\frac{x_i}{x_{i+1}-x^3_{i+1}}\ge \frac{n^3}{n^2-1}.\]here $x_{n+1}=x_1.$

2009 Mediterranean Mathematics Olympiad, 4

Tags: inequalities , inequalities unsolved

Let $x,y,z$ be positive real numbers. Prove that \[ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} \] [i](Proposed by Šefket Arslanagić, Bosnia and Herzegovina)[/i]

1996 Vietnam National Olympiad, 3

Tags: algebra , polynomial , calculus , derivative , inequalities unsolved , inequalities

Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$ where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$

2008 Brazil National Olympiad, 3

Tags: calculus , inequalities , trigonometry , inequalities unsolved

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

2000 All-Russian Olympiad, 5

Tags: inequalities , trigonometry , inequalities unsolved

Prove the inequality \[ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. \]

1984 IMO Longlists, 5

Tags: inequalities unsolved , inequalities

For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. If $m \ge 3$, prove that \[\left[\frac{m(m+1)}{2(2m-1)}\right]=\left[\frac{m+1}{4}\right]\]

2006 China Northern MO, 8

Tags: inequalities , inequalities unsolved

Given a sequence $\{ a_{n}\}$ such that $a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}$ , $n \in N$, $a_{0}=\frac{1}{2}$. Prove that $1-\frac{1}{2008}< a_{2006}< 1$.

1978 IMO Longlists, 44

Tags: trigonometry , inequalities , inequalities unsolved

In $ABC$ with $\angle C = 60^{\circ}$, prove that \[\frac{c}{a} + \frac{c}{b} \ge2.\]